"At some point, the amount of oil found will not keep up with the amount being used"
You have given no reason to think this must be the case. As more oil is used and more money is made in the industry, more resources are mobilized to finding additional oil. Techniques are improved, searches extended, theoretical understanding increases, etc.
As an energy budget matter, all human energy use is still only a fraction of known, lasting sources of input power (sun and radioactive decays and core cooling etc). The form that takes (mere heat, chemically stored, biomass, the exact mix etc) is an unknown.
And incidentally, the peak oil theory is a claim about absolute production, not production relative to use. And there is no sign to date of oil production declining. It is higher now than it has ever been. Those promoting it rest everything on distinctions among field sizes, which is an inherently murky measure, since it is essential an arbitrary net of distinctions thrown over an amorphous blob of data (are these ten wells in one field or in ten or in five?)
Naturally the easiest stuff to find is found before harder stuff to find. But there is no a priori reason to think the unfound stuff has to be a smaller volume overall. It is by definition unfound.
As long as the economic incentives are there, that will continue until eventually, there will be no place left to drill we have not.
Improvement in efficiency just means that more marginal resources become exploitable, but that does not 'make' more oil, it just makes some available which would not have been otherwise.
That comes at a cost, and economic factors apply, but at some point, even the most efficient extraction methods will have extracted all that can be removed from the reservoir.
As for reason to think that eventually we will not be able to keep up with the depletion of known reservoirs by discovering new ones?
Yep I do have reason to believe this. I have been working in the industry for 26 years, and recognize that oilfields we have come to take for granted since the '50s either have depleted or are beginning to decline.
There is a reason OPEC, especially Saudi Arabia, cannot drop the price of oil to the point exploration is less lucrative--they no longer have the excess production capacity to control the price that well, especially in the face of increasing global demand. They can create a shortage, but not a glut.
You note increased understanding as one of the keys to avoiding an eventual shortage, but that only comes with experience. Apparently, you are unfamilliar with the industry, which is notorious for boom and bust cycles which have kept tremendous numbers of people scurrying for the security of a steadier employment market. THere are a few of us die-hards who stick it through, but far, far more who do not. Especially in times like these, experienced personnel are at a premium.
Once someone is ensconced in a secure position, home every night, with only a 40 hour workweek, they probably are not coming back to the oil patch, especially if they still feel the sting of the layoffs that came with the last bust cycle.
Last, there are only a finite number of hydrogen and carbon atoms to string into useful chains, and I doubt that natural processes are outstripping our ability to break those molecules up.
Note, I am not addressing when, per se, this 'peak' will occur, only that it will, and the greater the demand for petroleum, the sooner it will happen.